The Metric Coalescent Process joint with David Aldous Daniel Lanoue - PowerPoint PPT Presentation

Introduction The Metric Coalescent The Metric Coalescent Process joint with David Aldous Daniel Lanoue June 17, 2014 Daniel Lanoue The Metric Coalescent Process

Introduction FMIE Processes The Metric Coalescent The Compulsive Gambler Two Related Processes Two stochastic processes: 1 The Compulsive Gambler Finite agent based model, 1 Finite Markov Information Exchange (FMIE) framework. 2 2 Metric Coalescent Measure-valued Markov process, 1 Defined for any metric space ( S , d ). 2 Daniel Lanoue The Metric Coalescent Process

Introduction FMIE Processes The Metric Coalescent The Compulsive Gambler FMIE Processes General Setup: interacting particle systems reinterpreted as stochastic social dynamics. 1 n agents; each in some state X i ( t ) ∈ S for each time t ≥ 0 2 Each pair of agents ( i , j ) meet at the times of a Poisson process of rate ν ij 3 At meeting times t between pairs of agents ( i , j ), the states transition ( X i ( t − ) , X j ( t − )) → ( X i ( t ) , X j ( t )) according to some deterministic or random rule F : S × S → S × S . Daniel Lanoue The Metric Coalescent Process

Introduction FMIE Processes The Metric Coalescent The Compulsive Gambler FMIE Processes Some familiar (and less familiar) examples: 1 Stochastic epidemic models; SIR model, etc. 2 Density dependent Markov chains (For ex. Kurtz 1978) 3 Averaging process, take S = R as money. Upon meeting two agents average their money. (Aldous-Lanoue 2012). � a + b , a + b � F ( a , b ) = 2 2 4 The iPod Model, an FMIE variant of the Voter Model (Aldous-Lanoue 2013) The goal is to study how the (non-asymptotic) behaviour depends on the finite meeting rates ν ij . Analogous to the study of mixing time for finite Markov chains. Daniel Lanoue The Metric Coalescent Process

Introduction FMIE Processes The Metric Coalescent The Compulsive Gambler Compulsive Gambler Process Simple FMIE process with agents’ state space S = R ≥ 0 , interpreted as money. When agents i and j meet they play a fair, winner take all game. I.e. the transition function is � a ( a + b , 0) with prob. a + b F ( a , b ) = b (0 , a + b ) with prob. a + b In the finite agent setting, we assume the total initial wealth is normalized � X i (0) = 1 . i ∈ Agents Importantly this allows us to view the state of the process as a probability measure of the set of agents. Daniel Lanoue The Metric Coalescent Process

Introduction FMIE Processes The Metric Coalescent The Compulsive Gambler Compulsive Gambler Process This model first studied in the setting of d -regular graphs and Galton-Watson Trees by Aldous-Salez. Some results on proportion of agents ”still alive” at a time t > 0, in particular t = ∞ [ALS14]. The rest of today’s talk will focus on a very particular variant of the CG, one with dependent rates ν ij . Daniel Lanoue The Metric Coalescent Process

Introduction The Finite Support Process The Metric Coalescent Generalizing to P ( S ) Extending the CG Process We can reformulate the CG as a measure-valued Markov process in terms of: 1 A metric space ( S , d ), 2 A rate function φ ( x ): R > 0 → R > 0 , The Metric Coalescent (MC) is then a continuous time P fs ( S )-valued Markov process, generalizing the CG as follows. For any µ ∈ P fs ( S ): 1 The atoms s i , 1 ≤ i ≤ # µ of µ are identified as the agents, 2 The masses µ ( s i ) as their respective current wealth, 3 The meeting rates between agents i and j given by φ and the metric as ν ij = φ ( d ( s i , s j )) Daniel Lanoue The Metric Coalescent Process

Introduction The Finite Support Process The Metric Coalescent Generalizing to P ( S ) A Visualization A simulation of the Metric Coalescent process on S = [0 , 1] 2 started from finitely supported approximations of the uniform measure: Link 1 Developed by Weijian Han. Daniel Lanoue The Metric Coalescent Process

Introduction The Finite Support Process The Metric Coalescent Generalizing to P ( S ) Main Theorem Let ( S , d ) be a locally compact, separable metric space and let the rate function φ ( x ) satisfy lim x ↓ 0 φ ( x ) = ∞ . Our main result for the Metric Coalescent is as follows [Lan14]: Main Theorem There exists a unique, cadlag, Feller continuous P ( S )-valued Markov process µ t , t ≥ 0 defined from any initial measure µ 0 ∈ P ( S ) s.t. if µ 0 is compactly supported: 1 µ t ∈ P fs ( S ) for all t > 0, almost surely; 2 For each t 0 > 0, the process ( µ t , t ≥ t 0 ) is distributed as the Metric Coalescent started at µ t 0 ; Daniel Lanoue The Metric Coalescent Process

Introduction The Finite Support Process The Metric Coalescent Generalizing to P ( S ) Proof Idea: Naive Approach The “naive” proof idea for constructing µ t , t ≥ 0 for a generic µ ∈ P ( S ) is to approximate µ with a sequence of finitely supported µ i ∈ P fs ( S ) for i ≥ 1. Then for t ≥ 0 define (the random measure) µ t as the weak limit i µ i µ t = lim t . Feller continuity in the Main Theorem retroactively implies that this sequence of random measures does converge, however – even ignoring the coupling issues here – this approach isn’t so fruitful in proving the Main Theorem. Some progress is made in [Lan14] following this idea using moment methods. Daniel Lanoue The Metric Coalescent Process

Introduction The Finite Support Process The Metric Coalescent Generalizing to P ( S ) Proof Idea: Exchangeable Coalescents Key Idea: replace the symmetric “random winners at meeting times” dynamics between agents with “deterministic winners according to a size-biased initial ranking”. This allows us to view the MC as an exchangeable partition process and enables a wide variety of tools. Among these used: 1 A comparison to Kingman’s Coalescent, 2 Two separate applications of de Finetti’s theorem, 3 An explicit formula for moments of � f d µ t for f : S → R . Daniel Lanoue The Metric Coalescent Process

Introduction The Finite Support Process The Metric Coalescent Generalizing to P ( S ) Further Directions Two directions for further research: 1 Coming Down From Infinity: We know that for compactly supported µ 0 initial measures, µ t is finitely supported for all positive times t > 0. It is easy to construct non-compactly supported µ 0 for which this isn’t true. What more can be said? 2 Time Reversal: A classical result on Kingman’s Coalescent is its duality under a time reversal to a conditioned Yule process. Viewing the MC as a “geometrization” of KC, can something similar be said? Daniel Lanoue The Metric Coalescent Process

Introduction The Finite Support Process The Metric Coalescent Generalizing to P ( S ) References Thanks for listening! For further information on these two processes and a complete reference list. [ALS14] D. Aldous, D. Lanoue, and J. Salez, The Compulsive Gambler Process , ArXiv e-prints (2014). [Lan14] D. Lanoue, The Metric Coalescent , ArXiv e-prints (2014). Daniel Lanoue The Metric Coalescent Process